Our model of four vdW layers in Fig. 1(a) contains the nonmagnetic conducting BLG with Bernal stacking [1]}, [2]} in the middle and two FM insulating layers located respectively on the upper and lower sides. As a typical conducting system, BLG has the advantages of highly-adjustable electron mobility [1]}, [4]} and long spin relaxation length [5]}, [6]}, [7]}, which can be very little affected by vdW proximity [8]}, [9]} and suggest an easily-prepared platform to realize low-consumption devices. We assume that the two insulating layers are identical 2D magnets with out-of-plane magnetization (favorable to ultrahigh density data storage [10]}), and do not contribute to the low-energy electronic bands except for proximity-induced magnetism. An effective low-energy Hamiltonian from an empirical tight-binding model (see Appendix A) reads
\(\begin{split}H_0=\upsilon _0(p_x\sigma _x+\xi p_y\sigma _y)-\frac{\gamma }{2}\sum _{i=x,y}\tau _i\sigma _i+m_z\tau _zs_z+\lambda _U\tau _z,\end{split}\) 
